Problem 37 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (10 \cdot 2)^{-2} $$

Step-by-Step Solution

Answer

The answer is \(\frac{1}{400}\).

Step 1: Simplify Inside the Parenthesis

Solve the multiplication inside the parenthesis first due to the order of operations (PEMDAS/BODMAS), which indicates Parenthesis, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). So, work out \(10 \cdot 2\) to get 20.

Step 2: Deal with Negative Exponentiation

Now, consider the exponentiation \(20^{-2}\). The negative exponent follows the rule that \(a^{-n} = \frac{1}{a^n}\). Consequently, \(20^{-2}\) becomes \(\frac{1}{20^2}\).

Step 3: Final Calculation

Finally, evaluate \(\frac{1}{20^2}\), which simplifies to \(\frac{1}{400}\).

Key Concepts

Order of Operations (PEMDAS/BODMAS)

When solving mathematical expressions, the order of operations is your best friend. This concept is critical to ensure everyone solves expressions the same way and gets the correct result. The acronym PEMDAS/BODMAS stands for the following sequence:

• Parentheses (Brackets)
• Exponents (Orders), which include powers and roots
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

For the expression \((10 \cdot 2)^{-2}\), the first operation involves solving inside the parentheses. Therefore, you calculate \(10 \cdot 2\), which equals 20. This ensures that the multiplication inside the parentheses occurs first, as dictated by the order of operations. It's essential to maintain this sequence to avoid mistakes.

Multiplication and Division

In math, multiplication and division are equally prioritized, meaning they are performed from left to right as they appear in the expression. In our example, when you see \(10 \cdot 2\), you immediately perform the multiplication step because it is within parentheses and is the innermost operation.
Once the expression inside the parentheses is simplified to 20, the next step would involve any division outside the parentheses, which is not the case here. It’s crucial to note this left-to-right rule in expressions where both operations are present together, as it helps prevent incorrect results. Understanding this hierarchy clarifies more complex expressions, allowing for accurate evaluations.

Evaluating Expressions

Evaluating expressions involves simplifying them to a single value, following the proper order of operations and rules such as those governing negative exponents. In the expression \((10 \cdot 2)^{-2}\), you begin by simplifying \(10 \cdot 2\) to 20. Then, apply the exponentiation rule: for any negative exponent \(a^{-n}\), the expression equals \(\frac{1}{a^n}\).
This means \(20^{-2}\) converts to \(\frac{1}{20^2}\). Evaluating \(20^2\) gives us 400. Thus, \(\frac{1}{20^2} = \frac{1}{400}\).
By understanding these steps, we correctly interpret and solve the expression while tackling the complexities involved in manipulating exponents and performing arithmetic operations.